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8    LEBESGUE'S CONVERGENCE THEOREMS       127

The usefulness of this theorem is that it does not require the convergence
to be uniform (not even on compact subsets). In particular:

(13.8.5)   Let (/) be a sequence of integrable functions^ such that the series with
general term    \fn\ djji is convergent. Then the series with general term fn(x) is

00

almost everywhere absolutely convergent in R, and if weputf(x) =  fn(x\
the function f (defined almost everywhere) is integrable and we have

(13.8.5.1)                   j/(x)^W = n|;J/B(x

(" term-by-term integration of a series of integrable functions").

N

For each integer N ^ 0, let #N =  \fn\. The sequence (#N) is an increasing

w=l

sequence of integrable functions, and

N   r

\L\ dn

((13.7.4) and   (13.7.3)).  By  (13.8.1), the function # = lim #N =  |/J  is

N->oo             n-1

integrable. This implies first of all that g is finite almost everywhere (13.6.4),
in other words that the series with general term \fn(x)\ is convergent almost
everywhere, which proves the first assertion ((5.3.1) and (5.3.2)). Moreover,

N

for all N we have ]T \fn\ ^ g, from which by (13.8.4) follow the integrability

n=l

of /and the relation (13.8.5.1).

(13.8.6) (i) Let E be a metric space, (x, z)\-^f(x, z) a mapping ofX x E into
K, and z0 a point ofE. Suppose that

(1)   for each z e E, the function x\-+f(x, z) is integrable ;

(2)   for almost all x e X, the function z\-^>f(x, z) is continuous at ZQ ;

(3)    there exists an integrable function g ^ 0 such that., for all z e E, we

have \f(x, z)\ ^ g(x) almost everywhere in X.
Then h(z) = \f(x, z) dfi(x) is continuous at ZQ .

(ii)   Suppose furthermore that E is an open interval in R and that f satisfies
the following conditions:

(4)   for almost all xeX, the function ZH->/(X, z) is finite and admits a
derivative D2/(x, z);re, for alln. Then f is integrable and